(1) Technical Field
The present invention generally relates to the field of controlling the shape of a deformable material, and more particularly to a method and apparatus for the optimal placement of actuators for controlling the shape of an elastically deformable material into a target shape using a number of actuators.
(2) Background
The problem of deformable material shape control and sensing has been studied most widely in the context of adaptive optics and segmented mirror telescopes using brute force approximations and sub-optimal over-engineering approaches for placement of and sensors and actuators. See, for example, G. Chanan, D. G. Mac Martin, J. Nelson, T. Mast, Control and Alignment of Segmented Mirror Telescopes: Matrices, Modes, and Error Propagation, Applied Optics, Vol. 43, No. 6, February 2004; and M. A. Van Dam, R. G. Lane, Extended Analysis of Curvature Sensing, J. Opt. Soc. Am. A 19, 13901397, 2002.
Conventional methods for shape sensing of deformable materials involve dense surface shape sensing. In order to achieve dense surface shape sensing, one has to resort to either dense range sensing or photogrammetric methods that reconstruct the three-dimensional (3-D) shape of a surface from stereoscopic views of the surface shape. These methods require high dimensional sensor information. Both range scanning and 3-D stereoscopic photogrammetry require costly equipment, dense sensing structures and computationally expensive sensor data processing stage.
A fundamental challenge in optimization of actuator placement for shaping deformable materials is the availability of efficient and simple models for computing the material shape as a function of geometric configuration of actuators. For example, classical convex optimization methods require computation of gradients corresponding to explicit mathematical description of the surface shape as related to geometry of the actuator network that in turn require a model to compute the gradients from and are often plagued by the local minima traps. These challenges so far have not been satisfactorily addressed in the scientific or industrial communities.
The ever increasing electromechanical complexity of actuation sensing and control mechanisms for automated systems in vehicles and flight systems makes any optimization technology a significant factor in energy cost saving ventures. For example, the reduction in number of actuators in flight systems and automobiles can lead to a significant decrease in production costs as well as energy savings measures. The reduction costs vary depending on the weight and power requirements of the actuation and control mechanisms which are often considerable.
For the foregoing reasons, there is a need for high precision shaping such that the discrepancy or mathematical error between the actual shape of an elastic material and that of the target shape is minimized. There is another need for an apparatus which determines the optimal placement of actuators responsible for the shaping of an elastically deformable material into target shapes using sparse number of actuators.